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A Posteriori Error Analysis of the Crank–Nicolson Finite Element Method for Parabolic Integro-Differential Equations

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Abstract

We study a posteriori error analysis for the space-time discretizations of linear parabolic integro-differential equation in a bounded convex polygonal or polyhedral domain. The piecewise linear finite element spaces are used for the space discretization, whereas the time discretization is based on the Crank–Nicolson method. The Ritz–Volterra reconstruction operator (IMA J Numer Anal 35:341–371, 2015), a generalization of elliptic reconstruction operator (SIAM J Numer Anal 41:1585–1594, 2003), is used in a crucial way to obtain optimal rate of convergence in space. Moreover, a quadratic (in time) space-time reconstruction operator is introduced to establish second order convergence in time. The proposed method uses nested finite element spaces and the standard energy technique to obtain optimal order error estimator in the \(L^{\infty }(L^2)\)-norm. Numerical experiments are performed to validate the optimality of the error estimators.

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Acknowledgements

The authors wish to thank both the referees for their valuable comments and suggestion which led to the improvement of this manuscript. G. Murali Mohan Reddy would like to thank FAPESP for the financial support received (Grant No. 2016/19648-9).

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Institute of Mathematics and Computer Sciences, University of São Paulo at São Carlos, P.O. Box 668, São Carlos, SP, 13560-970, Brazil

    G. Murali Mohan Reddy & José Alberto Cuminato

  2. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    Rajen Kumar Sinha

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  1. G. Murali Mohan Reddy
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  2. Rajen Kumar Sinha
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  3. José Alberto Cuminato
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Correspondence to G. Murali Mohan Reddy.

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Reddy, G.M.M., Sinha, R.K. & Cuminato, J.A. A Posteriori Error Analysis of the Crank–Nicolson Finite Element Method for Parabolic Integro-Differential Equations. J Sci Comput 79, 414–441 (2019). https://doi.org/10.1007/s10915-018-0860-1

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